Problem: The lifespans of porcupines in a particular zoo are normally distributed. The average porcupine lives $19.4$ years; the standard deviation is $3.3$ years. Use the empirical rule (68-95-99.7%) to estimate the probability of a porcupine living between $26$ and $29.3$ years.
$19.4$ $16.1$ $22.7$ $12.8$ $26$ $9.5$ $29.3$ $99.7\%$ $95\%$ $2.35\%$ $2.35\%$ We know the lifespans are normally distributed with an average lifespan of $19.4$ years. We know the standard deviation is $3.3$ years, so one standard deviation below the mean is $16.1$ years and one standard deviation above the mean is $22.7$ years. Two standard deviations below the mean is $12.8$ years and two standard deviations above the mean is $26$ years. Three standard deviations below the mean is $9.5$ years and three standard deviations above the mean is $29.3$ years. We are interested in the probability of a porcupine living between $26$ and $29.3$ years. The empirical rule (or the 68-95-99.7 rule) tells us that $99.7\%$ of the porcupines will have lifespans within 3 standard deviations of the average lifespan. It also tells us that $95\%$ of the porcupines will have lifespans within 2 standard deviations of the mean. That leaves $99.7\% - 95\% = 4.7\%$ of porcupines between 2 and 3 standard deviations of the mean, or $2.35\%$ on either side of the distribution. The probability of a particular porcupine living between $26$ and $29.3$ years is $\color{orange}{2.35\%}$.